Suppose that the sequence $(a_n)_{n \ge 1}$ of real numbers is such that $a_{n+1}− a_n \to 0$. Prove that the set of limits of its convergent subsequences is the interval with endpoints $\liminf_{n \to \infty} a_n$ and $\limsup_{n \to \infty}a_n$.
Can this be done this way that since the elements of the sequence are dense hence if you take a point then there are infinitely many elements around its neighbourhood and hence it acts as a subsequential limit now between lim sup and lim inf every point is a subsequential limit hence the proof follows Please help whether the above method is right or wrong